\begin{answer}{addnumbersgame}
The question about the wager is misleading, since it suggests that the outcome is random.
The crux of the question is to realise it is not.
It is easier to see this if you work backward.
If you want to win, you have to say ``$50$''.
You can only do this if the last number your opponent said was a number between $40$ and $49$.
That means, the number you have to say before that is $39$ (forcing your opponent to say a number in the range required).
Thus, saying $39$ will guarantee that you win.
You can work back further.
You can only say $39$ if the previous number you said was $28$ (forcing your opponent to say a number between $29$ to $38$, inclusive).
The number you'd have to say before that is $17$, and the first number you should say is $6$.
If you start and say ``$6$'', you will win with 100\% probability, and thus you should wager all the money at your disposal.
\end{answer}
